Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.00990

Galois Actions on Energy-Harmonic Functions

Bella Finkel
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the action of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on the solutions to the discrete Dirichlet problem for a graph with fixed edge energies. We establish transitivity of the Galois action outside a thin set in the parameter space and demonstrate that the exceptional thin set may be Zariski dense, so the conclusion cannot in general be replaced by one that holds on a Zariski-open set. Identifying the parameter locus where the Galois action is transitive constrains the possibilities for linking combinatorial and arithmetic properties of the discrete energy-harmonic Laplace equations and other systems of the same form.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Bella Finkel. 2026-10-31. Galois Actions on Energy-Harmonic Functions. https://arxiv.org/abs/2611.00990

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The virtual fundamental class for the moduli space of surfaces of general type

We prove that the moduli stack of index-one covers of semi-log-canonical surfaces of general type is isomorphic to the KSBA moduli stack of stable general type surfaces. Using the index-one covering Deligne-Mumford stack of a semi-log-canonical surface, we define the $\lci$ cover. The $\lci$ cover, as a Deligne-Mumford stack, has only locally complete intersection singularities. We then construct the moduli stack of $\lci$ covers so that it admits a proper map to the moduli stack of surfaces of general type. The smoothing of lci covering Deligne-Mumford stacks can not control all the non lci slc singularities. Thus, we construct the moduli stack of birational lci enhancement Deligne-Mumford stacks for all non lci slc singularities. This moduli stack is in general nonseparated. We construct a separated {\em algebraic space} from this moduli stack. Next, we construct a perfect obstruction theory on each of these moduli stacks and a virtual fundamental class in the Chow group. We then pushforward the virtual fundamental class to the KSBA moduli space. Thus, our construction proves Donaldson's conjecture on the existence of a virtual fundamental class for KSBA moduli spaces. A tautological invariant is defined by integrating a power of the first Chern class of the CM line bundle over the virtual fundamental class. This serves as a generalization of the tautological invariants defined by integrating tautological classes over the moduli space $\overline{M}_g$ of stable curves to the moduli space of stable surfaces.

math.AG↗
arXiv2026

$P=W$ via $\mathcal{H}_2$

Let $\mathcal{H}_2$ be the Lie algebra of polynomial Hamiltonian vector fields on the symplectic plane. Let $X$ be the moduli space of stable Higgs bundles of fixed relatively prime rank and degree, or more generally the moduli space of stable parabolic Higgs bundles of arbitrary rank and degree for a generic stability condition. Let $H^*(X)$ be the cohomology with rational coefficients. Using the operations of cup-product by tautological classes and Hecke correspondences we construct an action of $\mathcal{H}_2$ on $H^*(X)[x,y]$, where $x$ and $y$ are formal variables. We show that the perverse filtration on $H^*(X)$ coincides with the filtration canonically associated to $\mathfrak{sl}_2\subset \mathcal{H}_2$ and deduce the $P=W$ conjecture of de Cataldo-Hausel-Migliorini.

math.AG↗
arXiv2026

Fundamental groups, coregularity, and low dimensional klt Calabi-Yau pairs

In this article, we study how the absolute coregularity of a projective log pair reflects on its fundamental group. More precisely, we conjecture that for a projective klt log pair $(X,D)$ of absolute coregularity $c$ (and arbitrary dimension) the fundamental group $π_1^{\rm reg}(X,D)$ admits a normal abelian subgroup of finite index and rank at most $2c$. We prove this conjecture in the cases $0 \leq c \leq 3$, building on the almost abelianity of the fundamental groups of klt Calabi-Yau pairs of dimension $\leq 3$. In the cases $c \in \{0,1,2\}$ and fixed dimension, we can furthermore bound the index of a solvable normal subgroup. In dimension three, we are able to prove almost abelianity of the fundamental group of the regular locus for projective klt Calabi-Yau pairs.

math.AG↗